Such burners are preferably adapted for firing the combustion chamber of a gas turbine. A burner known from EP 1 389 714 A1 comprises a swirl generator and a downstream mixing tube. The burner head is provided with a pilot burner system to support the main flame. The pilot burner system comprises a mixing cavity for mixing pilot fuel and assist air. The mixing cavity of the known burner is of a pure circumferential shape. Within such a designed mixing cavity there is a risk for the pilot flame to burn inside.